#7018730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,25,31,37,29,41,32,39)(20,26,36,38,35,44,33,43)(30,40,45,47,34,42,46,48);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,33)(21,23)(25,39)(27,28)(30,45)(34,46)(35,36)(37,41)(38,43)(40,42);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,34)(21,32)(22,29)(23,31)(24,37)(26,42)(27,39)(28,41)(30,35)(33,45)(36,46)(38,48)(40,44)(43,47);(11,16)(13,15)(17,20,23,36,22,35,21,33)(18,26,24,38,28,44,27,43)(19,30,31,45,29,34,32,46)(25,40,37,47,41,42,39,48);(10,14)(13,15)(17,21,22,23)(18,27,28,24)(19,31,29,32)(20,33,35,36)(25,37,41,39)(26,43,44,38)(30,45,34,46)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6925926);